Secrets of Mental Math - Arthur Benjamin (bill gates best books .TXT) 📗

over until you don’t have to look back at the page, visualizing the turtle, Pancho, picking up your new mover, Ginger, as you do so. Got it?

Congratulations! You have just memorized the first twenty-four digits of the mathematical expression p (pi). Recall that p is the ratio of the circumference of a circle to its diameter, usually approximated in school as 3.14, or . In fact, p is an irrational number (one whose digits continue indefinitely with no repetition or pattern), and computers have been used to calculate p to billions of places.

THE PHONETIC CODE

I’m sure you’re wondering how “My turtle Pancho will, my love, pick up my new mover, Ginger” translates into 24 places of p.

To do this, you first need to memorize the phonetic code below, where each number between 0 and 9 is assigned a consonant sound.

1 is the t or d sound.

2 is the n sound.

3 is the m sound.

4 is the r sound.

5 is the l sound.

6 is the j, ch, or sh sound.

7 is the k or hard g sound.

8 is the f or v sound.

9 is the p or b sound.

0 is the z or s sound.

Memorizing this code isn’t as hard as it looks. For one thing, notice that in cases where more than one letter is associated with a number, they have similar pronunciations. For example, the k sound (as it appears in words like kite or cat) is similar to the hard g sound (as it appears in such words as goat). You can also rely on the following tricks to help you memorize the code:

1 A typewritten t or d has just 1 downstroke.

2 A typewritten n has 2 downstrokes.

3 A typewritten m has 3 downstrokes.

4 The number 4 ends in the letter r.

5 Shape your hand with 4 fingers up and the thumb at a 90-degree angle—that’s 5 fingers in an L shape.

6 A J looks sort of like a backward 6.

7 A K can be drawn by laying two 7s back to back.

8 A lowercase f written in cursive looks like an 8.

9 The number 9 looks like a backward p or an upside-down b.

0 The word zero begins with the letter z.

Or you can just remember the list in order, by thinking of the name Tony Marloshkovips!

Practice remembering this list. In about ten minutes you should have all the one-digit numbers associated with consonant sounds. Next, you can convert numbers into words by placing vowel sounds around or between the consonant sounds. For instance, the number 32 can become any of the following words: man, men, mine, mane, moon, many, money, menu, amen, omen, amino, mini, minnie, and so on. Notice that the word minnie is acceptable since the n sound is only used once.

The following words could not represent the number 32 because they use other consonant sounds: mourn, melon, mint. These words would be represented by the numbers 342, 352, and 321, respectively. The consonant sounds of h, w, and y can be added freely since they don’t appear on the list. Hence, 32 can also become human, woman, yeoman, or my honey.

The following list gives you a good idea of the types of words you can create using this phonetic code. Don’t feel obligated to memorize it—use it as inspiration to explore the possibilities on your own.

The Number-Word List

For practice, translate the following numbers into words, then check the correct translation below. When translating numbers into words, there are a variety of words that can be formed:

42
74
67
86
93
10
55
826
951
620
367

Here are some words that I came up with:

As an exercise, translate each of the following words into its unique number:

dog
oven
cart
fossil
banana
garage
pencil
Mudd
multiplication
Cleveland
Ohio

Answers:

Although a number can usually be converted into many words, a word can be translated only into a single number. This is an important property for our applications as it enables us to recall specific numbers.

Using this system you can translate any number or series of numbers (e.g., phone numbers, Social Security numbers, driver’s license numbers, the digits of p) into a word or a sentence. Here’s how the code works to translate the first twenty-four digits of p:

Remember that, with this phonetic code, g is a hard sound, as in grass, so a soft g (as in Ginger) sounds like j and is represented by a 6. Also, the word will is, phonetically, just L, and is represented by 5, since the consonant sound w can be used freely. Since this sentence can only be translated back to the twenty-four digits above, you have effectively memorized p to twenty-four digits!

There’s no limit to the number of numbers this code will allow you to memorize. For example, the following two sentences, when added to “My turtle Pancho will, my love, pick up my new mover, Ginger,” will allow you to memorize the first sixty digits of p:

And:

What the heck, let’s go for a hundred digits:

You can really feel proud of yourself once these sentences roll trippingly off your tongue, and you’re able to translate them quickly back into numbers. But you’ve got a ways to go for the world record. Hiroyuki Goto of Japan recited p to 42,195 places, from memory, in seventeen hours and twenty-one minutes in 1995.

HOW MNEMONICS MAKES MENTAL CALCULATION EASIER

Aside from improving your ability to memorize long sequences of numbers, mnemonics can be used to store partial results in the middle of a difficult mental calculation. For example, here’s how you can use mnemonics to help you square a three-digit number:

A Piece of Pi for Alexander Craig Aitken

Perhaps one of the most impressive feats of mental calculation was performed by a professor of mathematics at the University of Edinburgh, Alexander Craig Aitken (1895–1967), who not only learned the value of p to 1,000 places but, when asked to demonstrate his amazing memory during a lecture, promptly rattled off the first 250 digits of p. He was then challenged to skip ahead and begin at the 551st digit and continue for another 150 places. This he did successfully without a single error.

How did he do it? Aitken explained to his audience that “the secret, to my mind, is relaxation, the complete antithesis of concentration as usually understood.” Aitken’s technique was more auditory than usual. He arranged the numbers into chunks of fifty digits and memorized them in a sort of rhythm. With undaunted confidence he explained, “It would have been a reprehensibly useless feat had it not been so easy.”

That Aitken could memorize p to a thousand places does not qualify him as a lightning calculator. That he could easily multiply five-digit numbers against each other does. A mathematician named Thomas O’Beirne recalled Aitken at a desk calculator demonstration. “The salesman,” O’Beirne wrote, “said something like ‘We’ll now multiply 23,586 by 71,283.’ Aitken said right off, ‘And get …’ (whatever it was). The salesman was too intent on selling even to notice, but his manager, who was watching, did. When he saw Aitken was right, he nearly threw a fit (and so did I).”

Ironically, Aitken noted that when he bought a desk calculator for himself, his own mental skills deteriorated quickly. Anticipating what the future might hold, he lamented, “Mental calculators may, like the Tasmanian or the Maori, be doomed to extinction. Therefore you may be able to feel an almost anthropological interest in surveying a curious specimen, and some of my auditors here may be able to say in the year A.D. 2000, ‘Yes, I knew one such.’ ” Fortunately, history has proved him wrong!

As you recall from Chapter 3, to square 342 you first multiply 300 × 384, for 115,200, then add 422. But by the time you’ve computed 422 = 1,764, you may have forgotten the earlier number, 115,200. Here’s where our memory system comes to the rescue. To store the number 115,200, put 200 on your hand by raising two fingers, and convert 115 into a word like title. (By the way, I do not consider storing the 200 on fingers to be cheating. After all, what are hands for if not for holding on to digits!) Repeat the word title to yourself once or twice. That’s easier to remember than 115,200, especially after you start calculating 422. Once you’ve arrived at 1,764, you can add that to title 2, or 115,200, for a total of 116,964.

Here’s another:

After multiplying 300 × 246 = 73,800, convert 73 into gum and hold 800 on your hand by raising eight fingers. Once you’ve computed 272 = 729, just add that to gum 8, or 73,800, for a total of 74,529. This may seem a bit cumbersome at first, but with practice the conversion from numbers to words and back to numbers becomes almost second nature.

You have seen how easily two-digit numbers can be translated into simple words. Not all three-digit numbers can be translated so easily, but if you’re at a loss for a simple word to act as a mnemonic, an unusual word or a nonsense word will do. For example, if no simple word for 286 or 638 comes quickly to mind, use a combination word like no fudge or a nonsense word like jam-off. Even these unusual words should be easier to recall than 286 or 638 during a long calculation. For some of the huge problems in the next chapter, these memory tricks will be indispensable.

MEMORY MAGIC

Without using mnemonics, the average human memory (including mine) can hold only about seven or eight digits at a time. But once you’ve mastered the ability to change numbers into words, you can expand your memory capacity considerably. Have someone slowly call out sixteen digits while someone else writes them down on a blackboard or a piece of paper. Once they are written down, you repeat them back in the exact order they were given without looking at the board or the piece of paper! At a recent lecture demonstration, I was given the following series of numbers:

1, 2, 9, 7, 3, 6, 2, 7, 9, 3, 3, 2, 8, 2, 6, 1

As the numbers were called out, I used the phonetic code to turn them into words and then tied them together with a nonsensical story. In this case, 12 becomes tiny, 97 becomes book, 362 becomes machine, 793 becomes kaboom, 32 becomes moon, and 8261 becomes finished.

As the words were being created, I linked them together to form a silly story to help me remember them. I pictured finding a tiny book, which I placed inside a machine. This caused the machine to go kaboom, and tossed me to the moon, where I was finished. This story may sound bizarre, but the more ridiculous the story, the easier it is to remember—and besides, it’s more fun.

Chapter 8 The Tough Stuff Made Easy: Advanced Multiplication

Chapter 8

The Tough Stuff Made Easy:
Advanced Multiplication

At this point in the book—if you’ve gone through it chapter by chapter—you have learned to do mental addition, subtraction, multiplication, and division, as well as the art of guesstimation, pencil-and-paper mathemagics, and the phonetic code for number memory. This chapter is for serious, die-hard mathemagicians who want to stretch their minds to the limits of mental calculation. The problems in this chapter range from four-digit squares to the largest problem I perform publicly—the multiplication of two different five-digit numbers.

In